2,819 research outputs found
Clustering Partially Observed Graphs via Convex Optimization
This paper considers the problem of clustering a partially observed
unweighted graph---i.e., one where for some node pairs we know there is an edge
between them, for some others we know there is no edge, and for the remaining
we do not know whether or not there is an edge. We want to organize the nodes
into disjoint clusters so that there is relatively dense (observed)
connectivity within clusters, and sparse across clusters.
We take a novel yet natural approach to this problem, by focusing on finding
the clustering that minimizes the number of "disagreements"---i.e., the sum of
the number of (observed) missing edges within clusters, and (observed) present
edges across clusters. Our algorithm uses convex optimization; its basis is a
reduction of disagreement minimization to the problem of recovering an
(unknown) low-rank matrix and an (unknown) sparse matrix from their partially
observed sum. We evaluate the performance of our algorithm on the classical
Planted Partition/Stochastic Block Model. Our main theorem provides sufficient
conditions for the success of our algorithm as a function of the minimum
cluster size, edge density and observation probability; in particular, the
results characterize the tradeoff between the observation probability and the
edge density gap. When there are a constant number of clusters of equal size,
our results are optimal up to logarithmic factors.Comment: This is the final version published in Journal of Machine Learning
Research (JMLR). Partial results appeared in International Conference on
Machine Learning (ICML) 201
Three Puzzles on Mathematics, Computation, and Games
In this lecture I will talk about three mathematical puzzles involving
mathematics and computation that have preoccupied me over the years. The first
puzzle is to understand the amazing success of the simplex algorithm for linear
programming. The second puzzle is about errors made when votes are counted
during elections. The third puzzle is: are quantum computers possible?Comment: ICM 2018 plenary lecture, Rio de Janeiro, 36 pages, 7 Figure
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